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Brinkman's law as $Γ$-limit of compressible low Mach Navier-Stokes equations and application to randomly perforated domains

Peter Bella, Friederike Lemming, Roberta Marziani, Florian Oschmann

TL;DR

The paper establishes a Γ-type limit for time-dependent compressible Navier–Stokes equations in the low Mach regime on families of domains $Ω_ε$ that Mosco-converge to a fixed domain $Ω$, proving convergence to the incompressible Navier–Stokes–Brinkman system with a Brinkman term $μM\mathbf{u}$ when a strong limit solution exists. The authors develop a robust framework based on finite energy weak solutions, dissipative limits, energy inequalities, and a relative energy method to derive incompressibility, pass to the limit in the momentum equation, and prove weak–strong uniqueness. A key contribution is the treatment of the critical perforation regime ($α=3$) which introduces the Brinkman drag in the limit, extending prior stationary results to the time-dependent setting. They further apply the theory to randomly perforated domains, showing stochastic homogenization results where the holes are Poisson distributed with radii of order $ε^{3}$, and provide the necessary construction of test functions via Allaire-type corrections. The work provides a rigorous bridge between Brinkman-type drag in porous media and the low Mach limit of compressible flows, with implications for stochastic homogenization in random porous architectures.

Abstract

We consider the time-dependent compressible Navier-Stokes equations in the low Mach number regime inside a family of domains $(Ω_\varepsilon)_{\varepsilon > 0}$ in $\mathbb{R}^3$. Assuming that $\lim_{\varepsilon \to 0} Ω_\varepsilon = Ω\subset \mathbb{R}^3$ in a suitable sense, we show that in the limit the fluid flow inside $Ω$ is governed by the incompressible Navier-Stokes-Brinkman equations, provided the latter one admits a strong solution. The abstract convergence result is complemented with a stochastic homogenization result for randomly perforated domains in the critical regime.

Brinkman's law as $Γ$-limit of compressible low Mach Navier-Stokes equations and application to randomly perforated domains

TL;DR

The paper establishes a Γ-type limit for time-dependent compressible Navier–Stokes equations in the low Mach regime on families of domains that Mosco-converge to a fixed domain , proving convergence to the incompressible Navier–Stokes–Brinkman system with a Brinkman term when a strong limit solution exists. The authors develop a robust framework based on finite energy weak solutions, dissipative limits, energy inequalities, and a relative energy method to derive incompressibility, pass to the limit in the momentum equation, and prove weak–strong uniqueness. A key contribution is the treatment of the critical perforation regime () which introduces the Brinkman drag in the limit, extending prior stationary results to the time-dependent setting. They further apply the theory to randomly perforated domains, showing stochastic homogenization results where the holes are Poisson distributed with radii of order , and provide the necessary construction of test functions via Allaire-type corrections. The work provides a rigorous bridge between Brinkman-type drag in porous media and the low Mach limit of compressible flows, with implications for stochastic homogenization in random porous architectures.

Abstract

We consider the time-dependent compressible Navier-Stokes equations in the low Mach number regime inside a family of domains in . Assuming that in a suitable sense, we show that in the limit the fluid flow inside is governed by the incompressible Navier-Stokes-Brinkman equations, provided the latter one admits a strong solution. The abstract convergence result is complemented with a stochastic homogenization result for randomly perforated domains in the critical regime.
Paper Structure (24 sections, 16 theorems, 250 equations, 1 figure)

This paper contains 24 sections, 16 theorems, 250 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Notation.
  4. Setting of the problem and main result
  5. Suitable solution concepts
  6. Admissible domains and the main result
  7. Proof of Theorem \ref{['mainTheorem']}
  8. Step 1: Incompressibility.
  9. Step 2: Momentum Equation.
  10. Step 3: Energy inequality.
  11. Step 1: Relative Energy Inequality.
  12. Step 2: Weak-strong uniqueness.
  13. Application to randomly perforated domains
  14. General perforated domains and special test functions
  15. Validity of \ref{['H1']}--\ref{['H5']} in the random setting
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\varepsilon>0$ and $\Omega_\varepsilon, \Omega \subset D \subset \mathbb{R}^3$ be smoothly bounded domains confined to an overall bounded domain $D$. If $\lim_{\varepsilon \to 0} \Omega_\varepsilon = \Omega$ (in the sense of Definition def:omega_eps) and $\mathrm{Ma}(\varepsilon)$ with $\lim_{\

Figures (1)

  • Figure 1: Cells for the construction of $(\boldsymbol{\omega}_k^\varepsilon, q_k^\varepsilon)$.

Theorems & Definitions (39)

  • Theorem 1.1: See Theorem \ref{['mainTheorem']}
  • Definition 2.1: Finite energy weak solution to \ref{['NavierSystem']}
  • Remark 2.2
  • Definition 2.3: Dissipative solution to \ref{['systemstarkeloesung']}
  • Remark 2.4
  • Definition 2.5: Strong solution to \ref{['systemstarkeloesung']}
  • Definition 2.6: Well-prepared initial data
  • Definition 2.7: Assumptions on $(\Omega_{\varepsilon})_{\varepsilon>0}$
  • Theorem 2.8
  • Proposition 3.1: Convergence to dissipative solutions of \ref{['systemstarkeloesung']}
  • ...and 29 more